Coupled pair of one and two dimensional magneto-plasmons on electrons on helium
CCoupled pair of one and two dimensional magneto-plasmons on electrons on helium.
A.D. Chepelianskii, D. Papoular, H. Bouchiat, and K. Kono LPS, Univ. Paris-Sud, CNRS, UMR 8502, F-91405, Orsay, France LPTM, UMR 8089 CNRS & Univ. CergyPontoise, France ICST, NCTU, 1001 Ta Hsueh Rd., Hsinchu 300, Taiwan (Dated: October 18, 2019)Electrons on the liquid helium surface form an extremely clean two dimensional system wheredifferent plasmon-excitations can coexist. Under a magnetic field time reversal symmetry is brokenand all the bulk magneto-plasmons become gaped at frequencies below cyclotron resonance whilechiral one dimensional edge magneto-plasmons appear at the system perimeter. We theoreticallyshow that the presence of a homogeneous density gradient in the electron gas leads to the forma-tion of a delocalized magneto-plasmon mode in the same frequency range as the lowest frequencyedge-magnetoplasmon. We experimentally confirm its existence by measuring the correspondingresonance peak in frequency dependence of the admittance of the electron gas. This allows to re-alize a prototype system to investigate the coupling between a chiral one-dimensional mode anda single delocalized bulk mode. Such a model system can be important for the understanding oftransport properties of topological materials where states of different dimensionality can coexist.
The recent discovery of topological states of matterhas lead to striking predictions of topological surfacesand edge states [1–11]. However, it has so far been diffi-cult to realise a system where topological edge states arecompletely decoupled from remaining bulk states or spu-rious edge states of non-topological origin [10–13]. Thusunderstanding the interaction between topological edgemodes and non-topological bulk modes is highly impor-tant. Electrons on helium are a high purity two dimen-sional system where chiral edge magnetoplasmons modesnaturally form under a perpendicular magnetic field [14–17], interestingly their topological origin has been recog-nized only recently [18–20]. Bulk magneto-plasmon ina two dimensional electron gas have a gap at frequen-cies below the cyclotron resonance, and it is tradition-ally considered that edge-magnetoplasmons are the onlylow frequency plasmon excitations [21, 22]. In experi-ments with electrons on helium the frequency of edgemagnetoplasmons (EMP) is typically in the kHz range,while the cyclotron resonance frequency is typically sev-eral GHz. We show how a low energy bulk mode canbe created inside the bulk-magnetoplasmon gap by ananisotropic gradient of electronic density. In this letterwe will describe this plasma excitation as a magneto-gradient mode. We show that the frequency of thismagneto-gradient mode can be obtained from an effec-tive Schr¨odinger equation allowing to control the reso-nance frequency through the shape of the electron cloud.This allows to tune this frequency into resonance with theedge magnetoplasmons creating a model setting to studythe interaction between bulk and topological edge modes.We note that the existence of this low frequency bulkmagneto-plasmon can also be important to understandthe surprising collective effects that appear in electronson helium in the microwave induced resistance oscilla-tion regime [23–29]: zero-resistance states [30, 31] andincompressibility [32, 33] which are not yet understood microscopically.We first show that the presence of a density gradi-ent can indeed lead to the formation of a low frequencydelocalized magneto-plasmon, this may seem counter-intuitive as in an homogeneous system all the bulkmagneto-plasmons are gaped with their lowest frequencygiven by the cyclotron frequency ω c = eB/m . The equa-tions of motion for magnetoplasmons can be derived fromthe drift diffusion equations on the electronic density n e = n a + n t where we decompose the electronic densityinto time averaged and time dependent parts. Treatingthe time-dependent terms as a small perturbation, we canlinearise the drift diffusion equations which describe theelectron flow on the liquid helium surface (see a sketchof the cell geometry in Fig. 1): ∂ t n t = div d [ n a ( µ xx ∇ d V t + µ xy u z × ∇ d V t )] (1)Here V t is the time dependent part of the quasi-staticelectric potential V = V s + V t , in deriving this equa-tion we took into account that the electron cloud screensthe static-part of the electronic potential which leads to ∇ V s = 0, the longitudinal and Hall mobilities are givenby µ xx and µ xy . Experiments typically take place in thehigh magnetic field regime µ xx (cid:28) µ xy and µ xy (cid:39) B − (this corresponds to ω c much faster than the scatteringrate). Hence it is reasonable to first find the frequencyof the resonant plasmon modes in the limit µ xx = 0.The potential V t can be determined from the time de-pendent density n t by solving the electrostatic Poissonequation, for simplicity we will assume for now a localelectrostatics approximation n t = χV t where the com-pressibility χ = (cid:15) he is obtained from a plane capacitormodel. Simulations with an exact solution of electrostaticequations will be presented later. The static electron den-sity in presence of a density gradient can be written as n a ( r, θ ) = n ( r ) + n c ( r ) cos θ where ( r, θ ) are polar coor-dinates on the helium surface oriented along the gradient a r X i v : . [ c ond - m a t . m e s - h a ll ] O c t direction. Away from the edges of the electron gas, wecan approximate n c ( r ) = λr and treat the gradient λ as a small parameter anisotropy parameter. It is thusnatural to expand n t and V t in harmonics of the angle θ : n t = n t ( r ) + n tc ( r ) cos θ + n ts ( r ) sin θ (2)where we have kept the lowest harmonics. This proce-dure is justified since the only anisotropy comes from theuniform density gradient which couples only nearby har-monics through the cos θ term. Expanding the continu-ity equation to the lowest polar angle harmonics Eq. (1)we find an effective Schr¨odinger equation which describesstanding-modes of electron-density oscillations: χ ω µ xy ψ = − λ ∂ r ψ + 3 λ r ψ + ( ∂ r n ) r ψ (3)in this equation we introduced the effective wavefunction ψ ( r ) = √ rn ts and ω is the frequency of the density os-cillation, its time dependence obeys ∂ t ψ = − ω ψ . Thisequation describes a radial wave which propagates at ve-locity v = µ xy λ/ √ χ . As in quantum mechanics, theshape of the wavefunction ψ is controlled by the exter-nal potential. From Eq. (3) we see that it contains aterm describing repulsion at the origin and a confine-ment term proportional to the square of the gradient ofthe static isotropic density distribution ( ∂ r n ) . The ob-tained plasmon mode exists only due to the simultaneouspresence of a magnetic field and of the anisotropic den-sity gradient λ , we will thus call it a magneto-gradientplasmon (MGP). The frequency, ω g , of first MGP modeis given by the ground state of the Schr¨odinger equa-tion Eq. (3), it scales as ω g ∼ µ xy λ/ ( χR ) where R isthe radius of the electron cloud. This frequency vanishesthe limit R → ∞ , its behaviour is thus similar to EMPwhich also do not have a gap and can have frequenciesmuch below ω c . The MGP frequency drops to zero at λ = 0 and will be overdamped if the density gradientsare not strong enough. Fortunately due to the high mo-bilities of the electrons on helium system this mode canbe visible even for small density gradients. Since Eq. (3)is a standing wave equation, in addition to the lowestfrequency mode ω g resonances are expected around itsharmonics ω n = nω g ( n ≥ n t = χV t , and doesnot provide a complete description of the low frequencymagneto-plasmon modes. Indeed Eq. (3) does not predictany finite frequency modes in the limit λ → n ( r ) -1.0 -0.5 0.0 0.5 1.0 radius r ( cm ) liq. Heelectron cloud V d V d + V b n e “plateau”“caldera” V b h = . mm (a)(b) V d + V b V b V b < 0V b > 0 α V ac V ac Y(f ac ) centerguard guard Δn e (V b <0) Δn e (V b >0) xy FIG. 1. (a) Sketch of the experimental cell with applied DCand AC voltages. The helium cell and electrodes have cylin-drical symmetry but the helium level is slightly tilted with anangle α . A trapping potential V d = 7 V is applied to all bot-tom electrodes and a bias voltage V b can adjust the densitybetween the outer guard and central regions. The cell admit-tance Y ( f ac ) at frequency f ac = ω/ (2 π ) is measured betweento central and middle electrodes on which an AC potential V ac at frequency f ac is applied. Panel (b) shows the typical“plateau” and “caldera” profiles which occur respectively at V b < V b >
0. Orange and purple arrows illustrate thetwo possible definitions of ∆ n e in Eq. (4) for the two types ofdensity profile. a more realistic theory, reproducing the already knownmagnetoplasmons is thus needed. Such a theory has togo beyond the local density approximation and treat thelong range Coulomb interactions in realistic way. Thisrequires to fix the electrostatic environment of the elec-tron gas and its properties. From here we will focus on arealistic model of our experimental setup with electronson helium.A sketch of the system is shown on Fig. 1, electrons aretrapped on a helium surface by a pressing electric field.If the pressing electric field is perfectly perpendicular tothe helium surface the geometry has cylindrical symme-try with respect to the polar angle θ and no gradient ispresent λ = 0. However a small misalignment angle α between the electric field direction and the normal of thehelium surface leads to an in plane electric field compo-nent αE ⊥ ( α (cid:28)
1) which will create, within the localdensity approximation, a density gradient λ = χαE ⊥ .Since we assume that the helium surface remains flat inthe region where electrons are confined, the drift diffusionequation Eq. (1) at the surface remain unchanged for fi-nite α and it is only the relation between n t and V t whichis changed when we use the exact non-local electrostatics.Since the normal of the electric field presents a discon-tinuity as it crosses the electrons on helium cloud, a di-rect perturbation theory expansion around the isotropic (a) Experiment (b) Theory Re Y(f ac )[a.u.] f ac ( k H z ) R e n t [a . u . ] V b (Volt) V b (Volt) f ac = . ( k H z ) f ac = ( k H z ) (c) Theortical n t profies x (cm) x (cm) y ( c m ) y ( c m ) FIG. 2. (a) Magnetoplasmon modes appear as peaks in thereal part of the cell admittance Y ( f ac ) which is shown hereas function of the density profile (controlled by the bias V b between guard and central regions) and excitation frequency f ac . Two magneto-plasmon modes are observed in the ex-plored frequency range with very different dependence on V b :a dispersing mode at higher frequency and a low frequencymode with very small dependence of the electron cloud den-sity profile. (b) Finite element simulations of Y ( f ac ) based onEq. 1 taking into account a small tilt α = 0 . n t , the low frequency “magneto-gradient” plas-mon (polygon symbol) is delocalised across all the electroncloud, whereas the higher frequency plasmon (star symbol) isan edge magneto-plasmon (or inter-edge at V b >
0) propagat-ing in one-dimension, the two modes seem to make an avoidedcrossing when their frequencies overlap near V b = 0. solution is not possible. We derived a suitable pertur-bation theory expansion by performing a transformationinto a curved set of coordinates where the position of theinterface remains fixed with α . This expansion, to thelowest order, leads to a modified Laplace equation whichis given in Appendix. Finite elements (FEM) [34] sim-ulations based on this equation, confirm the validity ofthe approximation n c ( r ) (cid:39) αχE ⊥ r for all the shapes ofthe electron cloud explored experimentally except nearthe edge of the electron cloud. Thus a small inclinationof the helium surface with respect to the helium cell cre-ates a well defined density gradient which only weaklydepends on the shape of the electron cloud n ( r ). Tosimulate approximately the AC response of electrons onhelium, we used the Poisson equation on AC potential inthe limit α = 0 and drift diffusion equations on the he-lium surface to determine the expected admittance of thecell. The full equations and a discussion on their formalvalidity are provided in Appendix.We now present experiments that reveal the co-existence of 1D EMP and 2D MGP plasmons, the sim-ulations, which are presented simultaneously, will al- low to confirm the identification of the observed modes.The experiments were realised on an electron cloud (seeFig. 1) with N e = 3 × electrons at a magnetic fieldof B z = 0 . V ac with a 30mV amplitudeis applied on the intermediate top-ring electrode at afrequency f ac (1 to 10 kHz) and the induced pick-upsignal from the top central electrode is then measuredwith a voltage amplifier and a lock-in detector giving theAC cell admittance Y ( f ac ). The position of magneto-plasmon resonances then shows as peaks in the admit-tance Y ( f ac ) for a fixed bias voltage V b . This bias be-tween the outer guard and central electrodes can tunethe frequency of the magneto-plasmons by controlling theshape of the electron cloud [32, 35]. For V b < n ( r ) is a monotonously decaying function of theradial distance to the cloud center r , while for V b > v EMP (cid:39) ∆ n e π(cid:15) B ln qh (4)where q is the wave vector ( q = 1 /R for the lowest fre-quency mode where R is the radius of central bottomelectrode) and ∆ n e the difference in electron density be-tween the electron density in the center and guard regions(this definition is discussed in more precisely below).For the magneto-gradient plasmon the propagation ve-locity is given by: v MGP (cid:39) αE ⊥ B √ λ = αχE ⊥ . It depends on the perpendicu-lar electric field E ⊥ but not on ∆ n e as opposed to theEMP modes. This difference in the ∆ n e dependence pro-vides a convenient method to distinguish between MGPand EMP modes. We represent the admittance Y ( f ac )as function of both V b and f ac as a color scale map thatallows to visualize the dependence of the mode frequencyon the voltage V b (and thus on ∆ n e ). Modes that “dis-perse” as function of V b are candidate 1D EMP modeswhile the absence of a V b dependence suggests a magneto-gradient mode.Our experimental results are shown on Fig. 2 togetherwith FEM simulations of the perturbation theory thatwe introduced. Both experiment and simulations showthe presence of two resonant modes at low frequency.A mode whose frequency that strongly depends on V b with a dependence that reminds the dispersion relationof “Dirac” fermions, and a second mode whose frequencyis almost independent on V b except when its frequencycrosses the frequency of the dispersive mode.To identify the dispersive mode as an EMP we showthe theoretical frequency expected from Eq. (4). For the“calderia” geometry (positive V b , black lines), we set ∆ n e as the difference between the electron density on top ofthe rim in the guard region and the density in the cen-ter of the electron-cloud, it reproduces the experimentalEMP frequency without adjustable parameters. For theplateau geometry (red line for negative V b ) the density inthe guard entering in ∆ n e had to be reduced by 20% com-pared to its maximum density in the guard. This phe-nomenological correction probably reflects a more com-plex situation where the boundary of the electron cloudboundary moves with V b as the cloud is pushed towardsthe center. FEM simulations predict the correct positionfor the EMP mode without adjustable parameters evenin this case. The predictions for the linewidth and ad-mittance amplitude are less accurate as they dependenton the ratio µ xx /µ xy which was assumed to be fixed to5 × − and without any density dependence. To confirmthe 1D character of this dispersive mode we also repre-sented the simulated oscillating density profile n t of theEMP mode on Fig. 2, the oscillating density is indeedlocalised in a narrow strip of width h at the boundarybetween the central and guard regions. Note that, theperturbation theory is not reliable at the outer edge ofthe electron cloud and the peak in n t on the outer cloudboundary may not be physical.The lowest frequency mode on Fig. 2 has a resonancefrequency which is independent of V b except near thecrossing points with the previously identified EMP mode.It is thus a candidate magneto-gradient plasmon (MGP)mode. To confirm this assignment, we checked that thisfrequency scales as ∝ E ⊥ /B as expected from Eq. (5).The only other parameter in Eq. (5) is the inclinationof the helium free surface α compared to the electricfield. It was not possible to control this angle preciselyin our experiment however we confirmed that this fre-quency changes indeed with a small variation (of around0.1 deg) of the fridge inclination (all other parameterswere kept fixed). The FEM simulations confirm the ex-istence of the low frequency magneto-gradient mode. Ascan be seen from its oscillating density profile on Fig. 2, this mode is indeed delocalised across the entire elec-tron cloud. The FEM simulations can be used to fix thetilt angle α , in Fig. 2 a good agreement far from thecrossing with the EMP mode is obtained for α = 0 . V g . This could be dueto the dependence of the mobility on the electron den-sity which is not taken into account in the model. IndeedMGP is delocalized and thus can be more sensible to mo-bility gradients which appear as a result of the densityvariations across the electron cloud. In the simulationswe fixed the ratio µ xx /µ xy to reproduce the linewidth ofthe EMP mode, the linewidth of the MGP is thus under-estimated compared to the experiment. 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Derivation of the effective Schr¨odinger equation Eq. (3)
This equation is derived from the drift diffusionequations written in the local density approximation: ∂ t n t n tc n ts = µ xy χr λ∂ r ( rn ts ) / n ts ∂ r n λr∂ r n t − n tc ∂ r n (6)The density gradient λ introduces a coupling between different angular harmonics of the electron density, introducingthe effective wavefunction ψ ( r ) = √ rn ts this quation can be reduced to the effective Schr¨odinger equation Eq. (3). Effective Poisson equation in deformed coordinates where the position of the helium layer is fixed.
To enable the use of standard perturbation series we thus need to perform a coordinate transformation which levelsthe helium surface in the cell while keeping in place the top and bottom electrodes. We choose a rotation in the ( x, z )plane (containing the electric field direction z and the slightly miss-aligned helium surface normal) with a heightdependent rotation angle φ ( z ) = α (cid:16) − z (cid:48) h (cid:17) . The transformation between coordinates is the realized by: (cid:18) xz (cid:19) = (cid:18) cos φ ( z (cid:48) ) x (cid:48) − sin φ ( z (cid:48) ) z (cid:48) sin φ ( z (cid:48) ) x (cid:48) + cos φ ( z (cid:48) ) z (cid:48) (cid:19) (7)where x (cid:48) , z (cid:48) are the new coordinates.In the new coordinates the Poisson equation, to first order in α , becomes:∆ V + 8 αh (cid:0) x∂ z V + 2 xz∂ z V − z∂ x V − z ∂ xz V (cid:1) = 0 (8)in this form it can be expanded in powers of α . We see from Eq. (8) that α induces a coupling only between neighboringangular harmonics, thus to first order in α to which we will limit ourselves here, only cos θ and sin θ terms will begenerated.We solved Eq. (8) for a stationary electron cloud without AC excitation, this allows us to find the static densityprofile n a ( r, θ ) = n ( r ) + αn c ( r ) cos θ induced by the tilt of the cell. In the stationary case the potential of theelectron cloud is constant and fixed by the total charge of the cloud. In the isotropic case the problem then reduces tofind a stable boundary of the electron could for which the electric field at the boundary vanishes, which was done in asystematic way for different geometries in [32]. To find the anistropic correction n c ( r ) we iterated Eq. (8) neglectingthe small deformation of the circular cloud boundary. While this approach should give accurate predictions in thecenter of the electron cloud, the validity of the perturbation theory breaks down near the cloud boundary. The resultsof our finite element calculations for n ( r ) and n c ( r ) are shown below on Fig.3. -2 0 2 4 6 8 10 12 14 16 18 20 0 0.2 0.4 0.6 0.8 1 1.2 1.4 n e ( c m - ) r (cm)N e = 4 x 10 , V d = 7 VV b = 2 VV b = 0 VV b = -2V -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 n sc ( c m - / r ad ) r (cm)N e = 4 x 10 , V d = 7 VV b = 2 VV b = 0 VV b = -2V χ δ V FIG. 3. Left hand side panel shows the evolution of the radial steady state density n ( r ) for different bias voltages V b betweencentral an guard reservoirs. The right hand side panel shows the tilted component of density n c under the same conditions(we remind that the total steady state density is the sum n ( r ) + αn c ( r ) cos θ ), n c depends only weakly on V b in contrast to n ( r ) and is well approximated by n oc = χαE ⊥ r (straight line). To be fully consistent we in principle need to use the modified Poisson equation Eq. (8) which introduces a mixingbetween harmonics due to the tilt, we found however that using the usual Poisson equation ∆ V = 0 gives alreadya good description of the experiment. This probably comes from the fact that the dynamic drift-diffusion equationsalready introduces a coupling between modes through the density gradient n c and many (but seemingly not all) ofthe terms that would come from Eq. (8) become second order in α . Full drift diffusion equations for the lowest angular harmonics
For reference we write the full drift diffusionequations as a function of the steady state radial distribution n and the density gradient n c including contributionsfrom both µ xx and µ xy , these are the equations which are solved to build Fig. 2 in the main text.The AC potential V t is decomposed into its lowest harmonics V t = V t ( r ) + V tc ( r ) cos θ + V ts ( r ) sin θ . ∂ t n t n tc n ts = µ xy r ∂ r ( n sc V ts ) V ts ∂ r n n sc ∂ r V t − V tc ∂ r n + µ xx r ∂ r (cid:2) r ( n ∂ r V t + n sc ∂ r V tc ) (cid:3) ∂ r [ r ( n ∂ r V tc + n sc ∂ r V t )] − n r V tc ∂ r [ r ( n ∂ r V ts )] − n r V ts